Problem: The sum of two numbers is $33$, and their difference is $13$. What are the two numbers?
Answer: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 33}$ ${x-y = 13}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 46 $ $ x = \dfrac{46}{2} $ ${x = 23}$ Now that you know ${x = 23}$ , plug it back into $ {x+y = 33}$ to find $y$ ${(23)}{ + y = 33}$ ${y = 10}$ You can also plug ${x = 23}$ into $ {x-y = 13}$ and get the same answer for $y$ ${(23)}{ - y = 13}$ ${y = 10}$ Therefore, the larger number is $23$, and the smaller number is $10$.